Resolve IP address from MAC address

I am working in a network with subnet mask 255.255.0.0 (i.e 65534 hosts). The network has IP assignment scheme with DHCP server. In case of system restart or lease time completion, the system's IP is renewed. How would I be able to get the IP address of a system whose MAC address is available to me? Because in case of such large network, it would be difficult to ping each host and then read its MAC from arp cache.

I have also tried to use 'arping' utility but its also giving me TIMEOUT even though the system is pingable using IP address. This might be occurring because the arping utility sends broadcast ICMP request and most of the systems in my network do not reply to broadcast pings.